2)a)Does a linear map of two linear independent vectors $\underline{x}$,$\underline{y} \in \mathbb{R}^m$ to two linear dependent vectors $\underline{u}$,$\underline{v} \in \mathbb{R}^n$ exist/is possible?
b)And if it is what are the consequences for $ker(L)$?

1)No, because if two vectors are dependent:
$\exists k \in \mathbb{R}$,
so that: $k\underline{x}=\underline{y}$
and a linear map to two is linear dependent vectors $\in \mathbb{R}^n$is only possible by two vectors $\in \mathbb{R}^m$.

I'd like to suggest the following improvements for your MathJax code: use \to or \rightarrow for an arrow $\to$, and use \dim for $\dim$, \ker for $\ker$, and \operatorname{im} for $\operatorname{im}$ (these expressions should be upright). Please see here for how to typeset common math expressions with LaTeX, and see here for how to use Markdown formatting.
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Zev ChonolesMay 28 '13 at 22:25